Geodesic Radius of Curvature Calculation Method

[manushanker20]

I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula.

$$\frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G}$$

where $$s$$ is the arc length parameter and $$E$$, $$G$$ are the coefficents of the first fundamental form.

Can you please tell me how to perfrom the $$\partial G/\partial S$$? Since $$G=r_v\cdot r_v$$ I am not sure how to derivate it with respect to arc length

Thanks!

 

[UltrafastPED]

You will first express each of your functions in terms of the arc length - re-parameterize them.

 

[manushanker20]

I am dealing with non-uniform rational b-splines surface and I don't know the parametric equation of the geodesic path. I just know a set of points on the geodesic then how to re-parameterize with arc length.

can I use $$\frac{d G}{d s}=G_u \frac{d u}{d s}+ G_v \frac{d v}{d s}$$